Impenetrable barriers in quantum mechanics

Abstract

We derive the expression $V(x)\,u(x) = c\,\delta (x - a) + v(x)\,u(x)$ (where $V(x)$ is the potential, $u(x)$ the wave function, $c$ a constant and $v(x)$ a finite potential function for $x \le a)$, which is present in the one-dimensional Schrödinger equation on the whole real line when we have an impenetrable barrier at $x \ge a$, that is, an infinite step potential there. By studying the solution of this equation, we identify, connect and discuss three different Hamiltonian operators that describe the barrier. We extend these results by constructing an infinite square-well potential from two impenetrable barriers.